UK school terminology can be very confusing, and often specific to particular schools and areas.
In an ordinary state-funded secondary school, the usual word is "teacher" (and "headteacher", increasingly rarely "headmaster" / "headmistress"), and so a tutor means someone paid to help a child privately. You do also hear private tutor, but the following are the same:
- George needs some help with his mathematics, so he has a tutor once a week.
- ..., so he has a private tutor once a week.
I believe primary school age terminology would be the same as that.
Terminology in any given school might vary. My secondary school also had a teachers called "tutors", but these were the teachers responsible for a class of children (checked the register for example.)
In UK non-state schools, also called independent schools and public schools ("public" meant not in a private house, didn't mean that it was paid for by the state), terminology is even more specific to the individual school. For example at Eton, one of the best-known public schools, has a scheme of "tutors" who look after small groups of children, but also hold tutorials with them. It refers to teachers, but historically they were always masters, which is what you'll see in older texts: physics master, English master. In such a place, I'd expect to hear private tutor.
Some UK universities, most famously Oxford, are organised on the tutorial system, in which subject tutors hold tutorials with a few (typically two or three, sometimes only one) students; typically there is a personal tutor responsible for the student throughout their entire time at university. In an environment like this, you would have to say
- I needed help with second-order differential equations, so I found a private tutor to help me for a term
- My calculus tutor is no good, so I had to find a private tutor to help me
- I tutor in maths (would assume private unless in university context, "maths"=UK, "math"=US.)
To say "I tutor privately" is correct but awkward. I've heard "I'm a private tutor", and "George has a private tutor for physics and violin" (might be one person or two), and also "We have George tutored privately for physicals and violin".